Problem: In the diagram, the square has a perimeter of $48$ and the triangle has a height of $48.$ If the square and the triangle have the same area, what is the value of $x?$ [asy]
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
draw((3,0)--(6,0)--(6,5)--cycle);
draw((5.8,0)--(5.8,.2)--(6,.2));
label("$x$",(4.5,0),S);
label("48",(6,2.5),E);
[/asy]
Answer: Since the perimeter of the square is $48,$ its side length is $48 \div 4 = 12.$

Since the side length of the square is $12,$ its area is $12 \times 12 = 144.$

The area of the triangle is $\frac{1}{2} \times 48 \times x = 24x.$

Since the area of the triangle equals the area of the square, then $24x=144$ or $x=\boxed{6}.$